Monte Carlo Integration COS 323 Acknowledgment: Tom Funkhouser

Slide courtesy of Peter Shirley Integration in 1D f(x) x=1 Slide courtesy of Peter Shirley

Slide courtesy of Peter Shirley We can approximate g(x) f(x) x=1 Slide courtesy of Peter Shirley

Slide courtesy of Peter Shirley Or we can average f(x) E(f(x)) x=1 Slide courtesy of Peter Shirley

Estimating the average f(x) E(f(x)) x1 xN Slide courtesy of Peter Shirley

Slide courtesy of Peter Shirley Other Domains f(x) < f >ab x=a x=b Slide courtesy of Peter Shirley

Benefits of Monte Carlo No “exponential explosion” in required number of samples with increase in dimension Resistance to badly-behaved functions

Variance E(f(x)) x1 xN

Variance Variance decreases as 1/N Error decreases as 1/sqrt(N) E(f(x)) x1 xN

Variance Problem: variance decreases with 1/N Increasing # samples removes noise slowly E(f(x)) x1 xN

Variance Reduction Techniques Problem: variance decreases with 1/N Increasing # samples removes noise slowly Variance reduction: Stratified sampling Importance sampling

Stratified Sampling Estimate subdomains separately Arvo Ek(f(x)) x1 xN

Stratified Sampling This is still unbiased Ek(f(x)) x1 xN

Stratified Sampling Less overall variance if less variance in subdomains Ek(f(x)) x1 xN

Importance Sampling Put more samples where f(x) is bigger E(f(x)) x1 xN

Importance Sampling This is still unbiased E(f(x)) for all N x1 xN

Importance Sampling Zero variance if p(x) ~ f(x) E(f(x)) Less variance with better importance sampling x1 xN

Generating Random Points Uniform distribution: Use pseudorandom number generator 1 Probability W

Pseudorandom Numbers Deterministic, but have statistical properties resembling true random numbers Common approach: each successive pseudorandom number is function of previous Linear congruential method: Choose constants carefully, e.g. a = 1664525 b = 1013904223 c = 232 – 1

Pseudorandom Numbers To get floating-point numbers in [0..1), divide integer numbers by c + 1 To get integers in range [u..v], divide by (c+1)/(v–u+1), truncate, and add u Better statistics than using modulo (v–u+1) Only works if u and v small compared to c

Generating Random Points Uniform distribution: Use pseudorandom number generator 1 Probability W

Importance Sampling Specific probability distribution: Function inversion Rejection Split into 2 slides: one for theta, one for phi.

Importance Sampling “Inversion method” Integrate f(x): Cumulative Distribution Function 1 Split into 2 slides: one for theta, one for phi.

Importance Sampling “Inversion method” Integrate f(x): Cumulative Distribution Function Invert CDF, apply to uniform random variable 1 Split into 2 slides: one for theta, one for phi.

Importance Sampling Specific probability distribution: Function inversion Rejection Split into 2 slides: one for theta, one for phi.

Generating Random Points “Rejection method” Generate random (x,y) pairs, y between 0 and max(f(x))

Generating Random Points “Rejection method” Generate random (x,y) pairs, y between 0 and max(f(x)) Keep only samples where y < f(x)

Monte Carlo in Computer Graphics

or, Solving Integral Equations for Fun and Profit

or, Ugly Equations, Pretty Pictures

Computer Graphics Pipeline Modeling Animation Rendering Lighting and Reflectance

Rendering Equation Surface Light Viewer [Kajiya 1986]

Rendering Equation This is an integral equation Hard to solve! Can’t solve this in closed form Simulate complex phenomena Heinrich

Rendering Equation This is an integral equation Hard to solve! Can’t solve this in closed form Simulate complex phenomena Jensen

Monte Carlo Integration f(x) Shirley

Monte Carlo Path Tracing Estimate integral for each pixel by random sampling

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Surface Eye Pixel x

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Light x’ Eye x Surface

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Herf

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Surface Light Eye w w’ x Surface

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Debevec

Monte Carlo Global Illumination Specular Surface Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Light Eye w w’ x Diffuse Surface

Monte Carlo Global Illumination Rendering = integration Antialiasing Soft shadows Indirect illumination Caustics Jensen

Challenge Rendering integrals are difficult to evaluate Multiple dimensions Discontinuities Partial occluders Highlights Caustics Drettakis

Challenge Rendering integrals are difficult to evaluate Multiple dimensions Discontinuities Partial occluders Highlights Caustics Jensen

Monte Carlo Path Tracing Big diffuse light source, 20 minutes Jensen

Monte Carlo Path Tracing 1000 paths/pixel Jensen

Monte Carlo Path Tracing Drawback: can be noisy unless lots of paths simulated 40 paths per pixel: Lawrence

Monte Carlo Path Tracing Drawback: can be noisy unless lots of paths simulated 1200 paths per pixel: Lawrence

Reducing Variance Observation: some paths more important (carry more energy) than others For example, shiny surfaces reflect more light in the ideal “mirror” direction Idea: put more samples where f(x) is bigger

Importance Sampling Idea: put more samples where f(x) is bigger

Effect of Importance Sampling Less noise at a given number of samples Equivalently, need to simulate fewer paths for some desired limit of noise Uniform random sampling Importance sampling